With an incomplete solution. October 26, 1999
Rachel has the following utility function u(x1,x2)=½ln(x1)+ln(x2).
1.1 Find Rachel's demand functions for x1 and x2
1.2 In the initial state Rachel had an income of m=100 and the prices were p1=p2=2. Now the price of commodity x1 went down to p'1=1.
Find the change in the quantity of x1 demanded, that occurred due to the price change (find the initial and final bundles), and decompose it to a change due to the income effect and a change due to the substitution effect.
Explain the distinction between the two effects with the help of a diagram.
The demand functions are x1=m/3p1 and x2=2m/3p2. Thus in the initial situation x1=300/18 and in the final one x'1=600/18. So the change is an increase of 300/18.
To decompose the change we find the budget line corresponding to the pivotal change. The prices correspond to the final situation and the income is m"=1×100/6+2×200/6=500/6. The optimal quantity of x1 is x"1=500/18. Thus 200/18 units of x1 account for the substitution effect and the rest, 100/18 for the income effect.
1.3 Sarah has the utility function u(x1,x2)=ln(x1)+x2. Consider the change in quantity of x1 demanded by Sarah, corresponding to the same price changes as with Rachel. What would be the income effect? Explain your answer.
In this case the income effect would be zero, as x1 is a neutral good, and its level of consumption is independent of the income (in the relevant region of prices and income).
1.4 Find the change in Rachel's (consumer's) surplus due to the price change.
The change in consumers' surplus, measured in units of x2 is
2 / |m/p2 |----- dp1 = [100/6]ln2 |3p1 / 1
Jack consumes bread and wine. The price of bread is p2 dollars per loaf, and of wine p1 per bottle.
2.1 Draw as diagram of Jack's budget set, his demanded bundle and the indiference curve that correspondes this bundle.
2.2 Assume now that the government taxes wine with t dollars per bottle. Draw the new budget line and the new demanded bundle.
2.3 The government now considers to revise its taxation policy and impose on Jack a tax of w dollars. This amount is fixed and replaces the tax on wine. Will the government be able to collect more taxes than with the previous taxation policy, assuming that Jack's utility is identical in both taxation schemes?
See Section 5.6 in the textbook for a solution. In addition, there may be situations that the indeference curves have a kink. In this case it could be that both taxation policies would lead to exactly the same outcome.
A consumer has the utility function
x1 if x1<1 U(x1,x2)= ln[x1]+ln[x2+1]+1 otherwise
Draw a map of the consumer's indiference curves. Draw the income offer curve for the prices p1=p2=1. Draw the Engel curves for both x1 and x2.
The indiference curves are vertical parallel lines in the region x1<1 and have a regular convex-preference shape otherwise. In this region the indiference curves intersect with both the horizontal and vertical axes.
The income offer curve goes horizontally from (0,0) until (1,0). From then on it is a straight line that satisfies the equation x1=x2+1.